How to write an optimization constraint related to neighbors of a node in a graph

Hello All,

I am trying to implement an SDP optimzation problem in CVX. The language is Matalb. My question is mainly related to Matlab coding.

Here is the optimization problem (from the paper, Randomized Gossip Algorithms, P. 10 Eq. 54):

In this problem, W, P, and P_{ij} are n-by-n matrices. Also, in this problem, E is a set of neighbors of a node i. Constraint P_{ij}=0 if \{i,j\}\not\in E means that P_{ij} is zero if nodes i and j are not neighbors.

Does anyone can help with how to implement this neighborhood relationship? Any help is greatly appreiciated.

For n=3, neighbors.xlsx can look like:

I wrote this piece of code:

agt = struct([]);
    neighbors = readcell('neighbors.xlsx');
    N = 2;
    for i = 1:N
      agt(i).neighbors = neighbors{i};
    end
    j = 1;
    for i = 1:N
        D =[i,j];
        if ~ismember(D,agt(i).neighbors)
            W(i,j)== 0;
        end
        j = j+1;
    end

Answer 1

[Matt J]

You can get the graph adjacency matrix A from the adjacency() command,

https://www.mathworks.com/help/matlab/ref/graph.adjacency.html

Then, the constraints are:

P>=0,
dot(1-A(:),P(:))=0;
sum(P.*A,2)=1;

[Nasim]

Thanks for your reply!

Can you please explain what A is here? and how dot(1-A(:),P(:))=0; is the implemetion of the second constraint in the third line?

[Matt J]

A is the adjacency matrix of the graph. I recommended that you generate it using the adjacency() command.

The second constraint is implied by the two constraints,

P>=0
dot(1-A(:),P(:))=0;

because Q=1-A is a matrix such that Q(i,j)=1 if and zero otherwise. Therefore dot(1-A(:),P(:))=0 is the same as . Because we have applied the additional constraint , the constraint follows.

[Nasim]

Thanks a lot!